Clark model in general situation

For a unitary operator the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter $\gamma, |\gamma|=1$\mathrm{.}Namelyallsuchunitaryperturbationsare$U_\gamma:=U+(\gamma-1) (., b_1){\mathcal H} b$,where$b\in\mathcal H, |b|=1, b_1=U^{-1} b, |\gamma|=1$\mathrm{.}For$|\gamma|<1$operators$U\gamma$ are contractions with one-dimensional defects. Restricting our attention on the non-trivial part of perturbation we assume that $b$ is cyclic for $U$. Then the operator $U_{\gamma }$, $\mathrm{\mid }\gamma \mathrm{\mid }<1$ is a completely non-unitary contraction, and thus unitarily equivalent to its functional model ${\mathcal{M}}_{\gamma }$, which is the compression of the multiplication by the independent variable $z$ onto the model space $\mathcal K_{\theta_\gamma}$,where$\theta_\gamma$ is the characteristic function of the contraction $U_{\gamma }$. The Clark operator ${\mathrm{\Phi }}_{\gamma }$ is a unitary operator intertwining $U_{\gamma },\mathrm{\mid }\gamma \mathrm{\mid }<1$ and its model ${\mathcal{M}}_{\gamma }$, $\mathcal M_\gamma \Phi_\gamma = \Phi_\gamma U_\gamma$\mathrm{.}Ifspectralmeasureof$U$ is purely singular (equivalently, ${\theta }_{\gamma }$ is inner), operator ${\mathrm{\Phi }}_{\gamma }$ was described from a slightly different point of view by D. Clark. When ${\theta }_{\gamma }$ is extreme point of the unit ball in $H^{\mathrm{\infty }}$ was treated by D. Sarason using the sub-Hardy spaces introduced by L. de Branges. We treat the general case and give a systematic presentation of the subject. We find a formula for the adjoint operator ${\mathrm{\Phi }}_{\gamma }^{\ast }$ which is represented by a singular integral operator, generalizing the normalized Cauchy transform studied by A. Poltoratskii. We present a “universal” representation that works for any transcription of the functional model. We then give the formulas adapted for the Sz.-Nagy–Foias and de Branges–Rovnyak transcriptions, and finally obtain the representation of ${\mathrm{\Phi }}_{\gamma }$.